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and the relations between them.'* Not all schools of mathe-
matical thought would subscribe to these opinions, but they
suggest at least one 'philosophy' responsible for the following
definition of cardinal numbers.
Note that the initial stage in the definition is the description
of 'same cardinal number', in the spirit of Couturat's opening
remarks; 'cardinal number* then arises phoenix-like from the
ashes of its 'sameness*. It is all a matter of relations between
concepts not explicitly denned.
Two sets are said to have the same cardinal number when
all the things in the sets can be paired off one-to-one. After
the pairing there are to be no unpaired things in either
Some examples will clarify this esoteric definition. It is one
of those trivially obvious and fecund nothings which are so
profound that they are overlooked for thousands of years. The
sets (as, y9 z)9 (a9 b, c) have the same cardinal number (we shall not
commit the blunder of saying 'Of course! Each contains three
letters*) because we can pair off the things a?, y, s in the first set
with those, <z, 6, c in the second as follows, x with a, y with b,
2 with c, and having done so, find that none remain unpaired in
either set. Obviously there are other ways for effecting the
pairing. Again, in a Christian community practising technical
monogamy, if twenty married couples sit down together to
dinner9 the set of husbands will have the same cardinal number
as the set of wives.
As another instance of this 'obvious' sameness, we recall
* L. Couturat, De Vinflni ma&ematique, Paris, 1896, p. 49. With
the caution that much of this work is now hopelessly out of date> it
can be recommended for its clarity to the general reader. An account
of the elements of Cantorism by a leading Polish expert which is
within the comprehension of anyone with a grade-school education
and a taste for abstract reasoning is the Legons sur Us nombres trans-
jSnis, by Waclaw SierpinsM, Paris, 1928. The preface by Borel
supplies the necessary danger signal. The above extract from
Couturat is of some historical interest in connexion with Hilbert's
programme. It anticipates by thirty years Hubert's statement of his
formalist creed.